A simple DC motor model is shown below. It consists of a rotating mass, an input torque and viscous friction.where J=10
The system transfer can be found as follows:
We wish to control the motor position (θ) and have a damping ratio (ζ) of 0.707
The open characteristics of the system, shown below, don't meet the design specification. First the system does not have a ζ of 0.707. Second, the output is an integral of the input (pole at the origin). We wanted the output to follow the input.
Substituting into the differential equation for the system gives
This can also be done using block diagrams and transfer functionswhere θ is the measured position, θd is the desired position and kp is the feedback gain.
Now simplify the transfer function
Note that when analyzing the closed loop system using root locus the characteristic equation is
We can find kp using either root locus or, if we know the desired closed-loop poles, we can find kp directly through algebra.
Given the closed loop characteristic equation
From all of the possible solutions choose kp that results in ζ=0.707. Do this using brute force by selecting a kp and finding the roots of the resulting quadratic equation. As you change kp the root locations are indicated with an x on the root locus plot.
If you know the form of the characteristic equation for the closed-loop system, you can often solve directly for kp. In our problem closed-loop characteristic equation is:
Compare this to the standard 2nd order form
For the two equations to be equal each terms in the two equations must match. Set ζ=0.707, match terms and solve:
Alternately if we know the desired close-loop pole locations we can solve for kp directly. The key is that you must know the exact value for the pole locations and those must lie on the root locus. The desired pole locations from the root-locus graph are: s= -0.025±-0.025 i. Therefore the desired closed-loop equation must be
Use the following simulation to verify the solution. Select a value for k and change the value for θdesired. The settling time and damping ratio for the system should match the calculate settling time Tsettling and ζ) match the simulated response.. Experiment with different value for k and verify that you calculated closed-loop poles (
In addition to tracking θdesired the system also rejects disturbances to the motor. Click and drag on the wheel. Note how the torque increases to resist the disturbance.
The final check in the design is to verify that the motor torque requirements don't exceed the physical limits of the system. Starting at 0° set θdesired to 90° (or some value that represents the maximum change in position) and observe the maximum torque required by the control system. If this value exceeds the physical limits of the system you may need to reduce kp, limit the maximum change in the system, or limit how fast the system is allowed to change (this technique is discussed later in this text).